Machine Learning and Neural Networks - page 73
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11.2 McNemar's Test for Pairwise Classifier Comparison (L11 Model Eval. Part 4)
11.2 McNemar's Test for Pairwise Classifier Comparison (L11 Model Eval. Part 4)
Let's now discuss the McNemar test, which is a test we can use to compare two models to each other. This test is different from the cross-validation method we discussed last week for model selection. Unlike cross-validation, the McNemar test does not involve tuning the model using the training set. Instead, it assumes that we already have an existing model, such as a machine learning classifier mentioned in published literature, which we have access to through a web app or GitHub. The goal is to compare our own model to this existing classifier based on their performance on a test set.
The McNemar test allows us to compare two classifiers based on their performance on a test set. It was introduced by Quinn McNemar in 1947 and is a nonparametric statistical test for paired comparisons. In this case, we have a categorical dependent variable with two categories representing the correctness of predictions (e.g., correct or incorrect), and a categorical independent variable with two related groups representing the two models being compared. The pairing is achieved by using the same test set for both models. To perform the test, we use a two-by-two confusion matrix, which is a special version of the confusion matrix specifically designed for the McNemar test.
The two-by-two confusion matrix for the McNemar test includes the counts of predictions made by both models. For example, let's consider the case where we have true class labels of 0 and 1. Model 1 predicts 1 and 2, while Model 2 predicts 1 and 1. In this case, the count "a" represents the predictions that both Model 1 and Model 2 got correct (e.g., predicting 1 when the true label is 1). The count "b" represents the predictions where Model 1 is correct and Model 2 is wrong, and count "c" represents the predictions where Model 1 is wrong and Model 2 is correct. By tabulating these counts for a given test dataset, we can construct the two-by-two confusion matrix.
Using the two-by-two confusion matrix, we can compute various metrics. For instance, we can compute the prediction accuracy for each model by dividing the counts "a" and "b" by the total number of examples in the test set. Additionally, we are interested in the cases where Model 1 and Model 2 differ, represented by the counts "b" and "c" in the confusion matrix. These cases indicate where one model made a correct prediction while the other made an incorrect prediction. To compare the performance of the two models, we can run the McNemar test.
The McNemar test follows a typical hypothesis testing procedure. We define a null hypothesis and an alternative hypothesis. The null hypothesis assumes that the performances of the two models are equal, while the alternative hypothesis suggests that their performances differ. We compute a chi-square test statistic, which approximates a chi-square distribution. The test statistic is calculated as (b - c)^2 / (b + c). Based on this test statistic, we compute a p-value, assuming the null hypothesis is true. The p-value represents the probability of observing the given test statistic or a more extreme value under the null hypothesis.
To determine whether to reject the null hypothesis, we compare the p-value to a chosen significance level (e.g., 0.05). If the p-value is smaller than the significance level, we reject the null hypothesis and conclude that the performances of the two models are not equal. Conversely, if the p-value is greater than the significance level, we fail to reject the null hypothesis and assume that the performances of the two models are equal.
The continuity correction was introduced to address the issue of the chi-square distribution being a continuous distribution while the data in the 2x2 contingency table is discrete. By subtracting 1 from the absolute difference in the numerator, the continuity correction helps to provide a better approximation of the p-value. However, it is important to note that the use of continuity correction is not always necessary and depends on the specific context and data.
Another consideration in McNemar's test is the use of the exact binomial test. The exact binomial test provides an alternative approach to determine the statistical significance of the McNemar's test when the sample size is small or when the assumptions of the chi-square test are not met. The exact binomial test directly calculates the probability of obtaining the observed data under the null hypothesis, without relying on the chi-square approximation.
Additionally, McNemar's test can be extended to handle more than two models or treatments. In such cases, the test can be performed using a generalized McNemar's test or an extension known as the Cochran-Mantel-Haenszel test. These extensions allow for the comparison of multiple models or treatments simultaneously, taking into account the dependencies between the different groups.
In summary, McNemar's test is a nonparametric statistical test used to compare the performance of two models or treatments based on paired categorical data. It provides a way to assess whether the differences in performance between the models are statistically significant. The test involves constructing a 2x2 contingency table and computing a test statistic, which can be compared to the chi-square distribution or evaluated using the exact binomial test. By conducting McNemar's test, researchers can gain insights into the relative performance of different models or treatments in various fields, including machine learning, medicine, and social sciences.
11.3 Multiple Pairwise Comparisons (L11 Model Eval. Part 4)
11.3 Multiple Pairwise Comparisons (L11 Model Eval. Part 4)
In the scenario where you have a model that you want to compare with multiple other models found in literature or on platforms like GitHub, conducting pairwise comparisons can become problematic. If you have k different models, performing a pairwise test on each pair would result in k times (k - 1) / 2 combinations, which can be a large number. This raises concerns about the error rate when conducting multiple hypothesis tests.
Typically, hypothesis tests are conducted at a significance level of 0.05 or smaller. This means that if the null hypothesis is true, there is a 5% chance of incorrectly rejecting it, leading to a 5% error rate. When multiple tests are performed, the error rate for falsely rejecting a null hypothesis can increase to r times alpha, where r is the number of tests conducted. In the worst case, the error rate can be up to r times alpha if all pairwise null hypotheses are true.
To address this issue, a common approach is to use a protected procedure, which involves a two-step process. The first step is an omnibus test, where you assess whether there is a significant difference in the performance of all models combined. The null hypothesis assumes that the classification accuracies of all models are equal, while the alternative hypothesis suggests that they are not equal.
If the null hypothesis is rejected in the omnibus test, indicating that there is a difference in model performances, you can proceed to the second step, which involves pairwise post hoc tests. However, it is crucial to make adjustments for multiple comparisons to control the error rate. One commonly used adjustment method is the Bonferroni method, where the significance level is divided by the number of comparisons.
For the pairwise tests, the McNemar test can be employed. It is important to note that while these statistical procedures are formal and provide valuable insights, in machine learning practice, it is not very common to perform such extensive comparisons. Typically, researchers report prediction accuracies or errors and rank the models based on their performance.
While the Cochran-Q test can be implemented in the "mlxtend" library for comparing multiple models, it is worth mentioning that using such procedures for multiple model comparisons is still relatively uncommon in the field of machine learning. However, if you find a situation where you want to compare multiple models and perform statistical tests, you can explore these options and refer to the lecture notes and relevant literature for more detailed information.
It is important to note that these topics are not covered extensively in this lecture to allow sufficient time to cover other essential concepts like feature selection.
One issue with multiple comparisons is the increased risk of type I errors, also known as false positives. When conducting multiple tests, there is a higher chance of incorrectly rejecting null hypotheses, leading to erroneous conclusions. To mitigate this, researchers often apply adjustment methods like Bonferroni correction or false discovery rate (FDR) control.
The Bonferroni correction is a conservative adjustment that divides the significance level (alpha) by the number of comparisons (k). This adjusted significance level, denoted as alpha prime (α'), ensures that the overall familywise error rate remains under control. By using Bonferroni correction, each individual pairwise test is conducted at an alpha/k level.
Another popular method is FDR control, which focuses on controlling the proportion of false discoveries among all rejections. Instead of reducing the significance level for each comparison, FDR control adjusts the p-values of individual tests to control the overall false discovery rate. This method allows for a less stringent adjustment compared to Bonferroni correction, which can be advantageous in situations where a large number of comparisons are involved.
While Bonferroni correction and FDR control are widely used, it's important to note that they have their limitations. Bonferroni correction can be overly conservative, potentially leading to increased chances of type II errors or false negatives. On the other hand, FDR control may have more power to detect true differences but can also increase the risk of false positives.
In the context of machine learning, it's worth considering whether the goal is to assess pairwise differences comprehensively or to identify the top-performing model(s). Conducting pairwise tests for all possible combinations may be computationally expensive and time-consuming. In practice, researchers often focus on ranking the models based on their performance metrics and identifying the top-performing models rather than conducting formal statistical tests.
It's also important to recognize that statistical testing is just one aspect of model comparison. Other factors such as interpretability, computational efficiency, domain relevance, and practical considerations should also be taken into account when evaluating and selecting models.
In conclusion, while multiple comparisons and statistical tests can provide valuable insights into model performance comparisons, their practical application in machine learning is less common. Researchers often rely on reporting prediction accuracies or errors, visual comparisons, and ranking models based on performance metrics. Understanding the underlying statistical concepts and potential issues with multiple comparisons remains essential for conducting rigorous research and interpreting results accurately.
11.4 Statistical Tests for Algorithm Comparison (L11 Model Eval. Part 4)
11.4 Statistical Tests for Algorithm Comparison (L11 Model Eval. Part 4)
In the previous videos, we discussed how statistical inference can be used to compare different models that have already been fitted to a given dataset. Now, we will explore statistical tests that enable us to compare algorithms. This means we can compare models that have been fitted with different training sets or training subsets. This discussion focuses on the application of statistical inference and various statistical tests for algorithm comparison.
There are several statistical tests available for comparing algorithms, each with its pros and cons. In the lecture notes, you can find a more detailed explanation of these tests, along with additional materials. Here, I will provide an overview of the tests and highlight some key points.
One common test is the McNemar's test, which is primarily used for comparing models rather than algorithms. It is worth mentioning because it has a low false positive rate and is computationally efficient. However, it is not specifically designed for algorithm comparison.
Another test is the difference in proportions test, which unfortunately has a high false positive rate. This test requires fitting multiple models, making it more suitable for algorithm comparison. However, it can be computationally expensive due to the need for multiple model fittings.
The K-fold cross-validated t-test is another method used for algorithm comparison. It provides a more accurate assessment but still has a slightly elevated false positive rate. Despite this drawback, it remains a useful test.
The paired t-test with repeated cross-validation is another approach that requires fitting multiple models. While it has a lower false positive rate than some other tests, it can still be computationally intensive due to the repeated model fitting.
A more advanced technique is the 5x2 cross-validated paired t-test, which exhibits a low false positive rate and slightly higher statistical power compared to McNemar's test. It offers a more robust approach to algorithm comparison. Additionally, there are newer approaches, such as the 5x2 cross-validated paired f-test, that provide further improvements.
In the lecture notes, I delve into more detail about these tests and other statistical approaches. Additionally, I have implemented most of these tests in MLA extend, a library that accompanies the lecture materials. You can find implementations of the McNemar's test, Cochrane's Q test, the resampled paired t-test (not recommended), K-fold cross-validated paired t-test, 5x2 cross-validated combined t-test, and more.
While statistical tests offer valuable insights, there are also computational or empirical approaches to algorithm comparison. In the next video, we will explore the motivation behind algorithm selection and comparison in real-world applications. For example, we may want to compare different algorithms to identify the best performer in scenarios such as developing an email application or creating a recommendation system for research articles.
In conclusion, there are several statistical tests available for comparing algorithms, each with its strengths and limitations. These tests can provide insights into the performance differences between algorithms fitted with different training sets. However, it is important to consider computational efficiency, false positive rates, and statistical power when selecting an appropriate test. Additionally, empirical approaches can complement statistical tests in algorithm comparison.
11.5 Nested CV for Algorithm Selection (L11 Model Eval. Part 4)
11.5 Nested CV for Algorithm Selection (L11 Model Eval. Part 4)
Alright, let's dive into the topic of computational algorithm selection. In this discussion, we will focus on a technique called nested cross-validation and explore some code examples in the upcoming video. Before we delve into nested cross-validation, let's quickly recap some key points we covered earlier.
Previously, we discussed the three-way holdout method as a means of model selection. Here's a brief summary of the process: we start by splitting our original dataset into a training set and a test set. Next, we further divide the training set into a smaller training set and a validation set. The training set is used, along with a machine learning algorithm and specific hyperparameter values, to train a model. By iterating through various hyperparameter settings, we obtain multiple models with their respective performances. Finally, we select the model with the highest performance, as measured on the validation set, and evaluate its final performance on the test set. It's important to include an independent test set to mitigate any selection bias introduced during model selection.
Now, let's revisit the concept using a different figure to aid our understanding. In this diagram, we can visualize three scenarios. In the first scenario, we evaluate a single model that is trained on the training set and tested on the test set without any model tuning. This approach is suitable when no model tuning is required.
In the second scenario, we evaluate multiple models on the same training and test sets. Each model is trained with different hyperparameter settings, and we select the best performing one based on the test set performance. However, using the test set multiple times for model selection can introduce selection bias, making this approach less desirable.
The third scenario corresponds to the three-way holdout method we discussed earlier. Multiple models are trained on the training set with different hyperparameter settings. The validation set is then used to select the best performing model, which is subsequently evaluated on the test set. This approach helps mitigate selection bias by using a separate validation set for model ranking.
While the three-way holdout method is effective, a better approach is k-fold cross-validation, which we covered in our previous discussion. This method divides the data into k folds, with each fold taking turns as the validation set while the rest serve as the training set. This approach is particularly beneficial when the dataset is limited in size. For larger datasets, three-way holdout can still be a viable option, especially in deep learning where dataset sizes are typically larger, and additional considerations like model convergence come into play.
Now, let's move forward and discuss nested cross-validation, which takes us a step further by comparing different algorithms. Suppose we want to compare algorithms like K-nearest neighbors, decision trees, gradient boosting, and random forests. Each algorithm will undergo hyperparameter tuning to select the best model. We introduce another loop to the cross-validation procedure, resulting in nested cross-validation. The outer loop is responsible for model evaluation, while the inner loop focuses on hyperparameter tuning. This two-step procedure makes nested cross-validation more complex than regular k-fold cross-validation, as it essentially comprises two nested k-fold cross-validations.
To understand this process better, let's walk through an illustration starting with our original dataset. Imagine we have an independent test set for final evaluation, but for now, our main training set will suffice. Similar to k-fold cross-validation, we iterate through a loop for a specified number of folds, let's say five in this case. In each iteration, the data is split into training folds and a test fold. However, instead of training the model solely on the training fold and evaluating it on the test fold, we proceed to the next step.
In the next step, we take one of the training folds, such as the one at the bottom of the diagram, and further divide it into a smaller training set and a validation set. The smaller training set is used to train different models with various hyperparameter settings, while the validation set is used to select the best performing model.
Once the inner loop completes for the current training fold, we have a selected model with its corresponding hyperparameter settings. We then evaluate this model on the test fold from the outer loop, which was not used during the inner loop.
The process continues for each fold in the outer loop. Each time, a different fold is held out as the test fold, while the remaining folds are used for training and validation. This ensures that every fold is used as both a test set and a validation set, and that each model is evaluated on a different set of data. The final performance of the model is determined by averaging the performance across all folds.
Nested cross-validation helps us compare different algorithms by providing a more robust estimate of their performance. By repeating the nested cross-validation process multiple times, we can obtain more reliable and stable performance estimates.
To summarize, nested cross-validation is a technique that combines the benefits of model selection and hyperparameter tuning. It allows us to compare different algorithms by evaluating their performance on multiple folds of data and selecting the best model based on nested iterations of cross-validation. This approach helps mitigate selection bias and provides a more accurate estimation of algorithm performance.
In the upcoming video, we will explore code examples to demonstrate how nested cross-validation is implemented in practice. Stay tuned for the next part of this series.
11.6 Nested CV for Algorithm Selection Code Example (L11 Model Eval. Part 4)
11.6 Nested CV for Algorithm Selection Code Example (L11 Model Eval. Part 4)
Alright, now that we have discussed the concept behind nested cross-validation, let's delve into a code example. This example will help you gain a better understanding of nested cross-validation from a computational perspective. Additionally, it will prove useful for your class projects when comparing algorithms.
Firstly, the code examples can be found on GitHub. I have uploaded them to our regular class repository under the name "l11_code." There are three notebooks available: "verbose_one," "verbose_two," and "compact." All three notebooks produce the same results, but they differ in their implementation approach.
In the "verbose_one" notebook, I have taken a more manual approach by using the stratified k-fold method manually. On the other hand, in the "verbose_two" notebook, I have utilized the cross_validate function. Finally, in the "compact" notebook, I have used cross_val_score. Each notebook provides different levels of information when analyzed. For now, I recommend starting with the "verbose_one" notebook since you are already familiar with the stratified k-fold object.
Before we proceed, it's worth mentioning that the implementation approach you choose doesn't significantly impact the results. However, I wouldn't recommend using the "compact" notebook as it provides less information about the hyperparameter sets. Later on, I can briefly show you how the hyperparameter sets look after we discuss the following content.
Now, let's examine the "verbose_one" notebook, which demonstrates the manual approach to nested cross-validation. In this notebook, you will find an illustration of nested cross-validation and how it functions. The process involves an outer loop that runs the inner loops. For each outer loop, the fold is split into training and test portions. The training portion is then passed to the inner loop, which performs hyperparameter tuning or model selection. This can be achieved using grid search, as we learned in the previous lecture.
In the notebook, you will find the necessary setup steps, such as importing required libraries and modules. These include grid search for model tuning in the inner loop, splitting the dataset using stratified k-fold cross-validation, pipeline, standard scalars, and classifiers that we want to compare. For the purposes of this example, we are using a smaller version of the "emnes" dataset, which consists of 5000 training examples to keep the computational feasibility intact. Additionally, 20% of the dataset is set aside as test data, allowing us to compare the performance of nested cross-validation with the test set performance.
Moving forward, we initialize the classifiers we will be using. The first classifier is a logistic regression classifier, specifically a multinomial logistic regression classifier. This classifier is also known as softmax regression in deep learning. Although we haven't covered it in this class, we will cover it in "Statistics 453." The reason behind using this classifier is to have a wider range of algorithms to compare. Furthermore, it is relatively fast compared to other algorithms. Another relatively fast classifier we consider is the support vector machine, specifically the linear one. By including these classifiers, we aim to compare various hyperparameter settings.
It's important to note that decision tree-based classifiers such as the decision tree itself and the random forest classifier do not require parameter scaling. Therefore, we only perform scaling for the other classifiers. To facilitate this, we use pipelines, which combine the standard scaler with the respective classifier. Thus, the pipeline can be seen as the classifier itself. For each classifier, we define a hyperparameter grid that we will search. These grids contain the parameters we want to tune for each classifier. For example, in logistic regression, we consider the regularization weight penalty and different regularization strengths. Nearest neighbors involve considering the number of nearest neighbors and the distance.
In the "verbose_one" notebook, after defining the classifiers and their respective hyperparameter grids, we move on to setting up the outer and inner loops for nested cross-validation.
The outer loop uses stratified k-fold cross-validation to split the dataset into training and test sets. It iterates over the folds and keeps track of the fold index, training indices, and test indices. For each fold, the training data is further split into training and validation sets for the inner loop.
The inner loop performs model selection or hyperparameter tuning using grid search. It iterates over the hyperparameter grid for each classifier and uses the training and validation sets from the outer loop to find the best hyperparameters for each combination. Grid search exhaustively searches the specified hyperparameter grid and evaluates the performance of each combination using cross-validation.
After the inner loop completes, the best hyperparameters for each classifier are recorded. Then, the performance of the selected models is evaluated on the test set, which was set aside at the beginning. The evaluation metrics such as accuracy, precision, recall, and F1 score are calculated for each classifier.
Finally, the results of the nested cross-validation and test set evaluation are displayed, allowing you to compare the performance of different classifiers and their hyperparameter settings.
It's important to note that the "verbose_two" and "compact" notebooks provide alternative implementations of nested cross-validation using the cross_validate function and cross_val_score function, respectively. These functions handle some of the cross-validation steps automatically, simplifying the code. However, they may provide less detailed information compared to the "verbose_one" notebook.
I hope this overview helps you understand the code examples and how nested cross-validation is implemented. Feel free to explore the notebooks and experiment with different datasets and classifiers to gain a better understanding of the concept.
12.0 Lecture Overview (L12 Model Eval 5: Performance Metrics)
12.0 Lecture Overview (L12 Model Eval 5: Performance Metrics)
Hello everyone,
I hope you all had a wonderful Thanksgiving break and were able to relax and recharge before the final weeks of the semester. While it's unfortunate that the semester is coming to an end, there's still much to look forward to, particularly your project presentations. I'm excited to see what you have built based on the content we covered in this class and witness the creativity and application of your machine learning knowledge.
In the upcoming two weeks, I have some plans for our remaining time together. This week, I'll be covering model evaluation and focusing on five performance and evaluation metrics. The goal is to broaden your perspective beyond just classification accuracy and errors. We will explore various metrics that can help evaluate machine learning models effectively. I don't anticipate this topic taking up much time, so if we have additional time, I'll also touch upon feature selection. I shared some self-study material on this topic earlier since I knew we might not have time to cover it extensively. I want to be mindful of the challenges posed by online learning and not overwhelm you with too many topics in a short period. I understand that you all have a lot on your plates, including Homework 3, due on December 4, and your project presentations in video format on December 6.
Regarding the project presentations, next week, I will create Canvas pages where you can embed your presentations. Additionally, I will set up a quiz format for voting on the project awards, including the most creative project, the best oral presentation, and the best visualization. These awards will be determined by your votes. I believe it will add an element of fun to the process. I'll organize everything for next week, which means there won't be any lectures. However, I highly recommend that everyone watches the project presentations. There will be points awarded for filling out surveys related to the presentations. It's also fair to watch each other's presentations since you all put in significant effort. We can have discussions and ask questions on Piazza or explore other platforms that allow for interaction. I'll consider the best way to facilitate this engagement.
Before we dive into today's lecture, I want to remind you about the course evaluations. Our department requested that you provide feedback on how the semester went for you. This year has been different due to the online format, so your insights would be valuable. I will post the links for the course evaluations on Canvas. It would be greatly appreciated if you could take the time to fill them out. However, I want to emphasize that there is no penalty if you choose not to complete them. It's merely a request to gather your feedback.
With that said, let's begin with Lecture 5 on performance and evaluation metrics. We've come a long way in model evaluation, starting with bias-variance decompositions to understand overfitting and underfitting. We explored the holdout method for data set division and its pitfalls, constructing confidence intervals using the normal approximation method, resampling techniques like repeated holdout and bootstrapping, and cross-validation for model selection. Last week, we discussed statistical tests for model and algorithm comparisons, including nested cross-validation. Today, our focus will be on evaluation metrics.
We'll start by discussing the confusion matrix, which differs from the McNemar confusion matrix covered last week. From the confusion matrix, we can derive metrics such as the false positive rate, true positive rate, and others, which will be useful when we delve into the receiver operating characteristic. Additionally, we'll explore precision, recall, F1 score, Matthews correlation coefficient, and balanced accuracy. The latter is particularly useful in cases where class imbalances exist within the dataset. Towards the end, we'll address extending binary metrics to multiclass settings, with the exception of balanced accuracy, which is already compatible with multiclass classification.
In the next video, we'll begin our discussion with the confusion matrix.
12.1 Confusion Matrix (L12 Model Eval 5: Performance Metrics)
12.1 Confusion Matrix (L12 Model Eval 5: Performance Metrics)
Let's start by discussing the confusion matrix and its significance. In the lecture, the speaker mentioned that they didn't prepare lecture notes due to various reasons like the busy end of the semester and having covered the topic in a Python machine learning book. They suggested referring to Chapter 6 of the book for more details.
The confusion matrix is a tool used to evaluate the performance of a machine learning classifier. It shows the comparison between the predicted class labels and the actual class labels in a supervised classification problem. The matrix helps us understand how well the classifier is performing and which class labels it tends to confuse.
The confusion matrix is typically represented in a two-by-two format, also known as a contingency matrix. It consists of four components: true positives (TP), false negatives (FN), false positives (FP), and true negatives (TN). The "positive" class refers to the class of interest that we want to predict, while the "negative" class refers to the other class.
The true positives (TP) are the instances that belong to the positive class and are correctly identified as such by the classifier. On the other hand, false negatives (FN) are instances from the positive class that are incorrectly predicted as negative.
Similarly, false positives (FP) are instances from the negative class that are incorrectly predicted as positive. Lastly, true negatives (TN) are instances from the negative class that are correctly identified as negative.
By analyzing these components, we can calculate various performance metrics. The lecture mentioned two common metrics: classification accuracy and classification error. Classification accuracy is computed by dividing the sum of true positives and true negatives by the total number of predictions. On the other hand, classification error is calculated as one minus the accuracy.
The speaker then introduced the breast cancer Wisconsin dataset, which contains information about breast cancer diagnoses. They explained that the dataset has various columns, including an ID number for each patient and features extracted from digitized images of cancer cell nuclei.
To prepare the dataset for classification, the speaker used a label encoder from scikit-learn to transform the string class labels (malignant and benign) into integer labels (0 and 1). They split the dataset into a training set (80%) and a test set (20%).
Next, the speaker demonstrated how to plot a confusion matrix using a k-nearest neighbor classifier. They emphasized the importance of feature scaling for KNN classifiers and mentioned the use of a standard scalar and pipeline for preprocessing.
To visualize the confusion matrix, the speaker used the confusion_matrix function from the mlxtend library. The resulting confusion matrix was displayed using matplotlib, with the true positives in the lower right corner and true negatives in the upper left corner.
Additionally, the speaker mentioned some optional parameters of the confusion_matrix function, such as show_absolute and show_normed. These parameters allow for customization of the visualization, including showing the absolute numbers or the normalized values.
Finally, the speaker discussed metrics derived from the confusion matrix, such as true positive rate, false positive rate, false negative rate, and true negative rate. These metrics are important for evaluating classifier performance and will be further explored in relation to the receiver operator characteristic (ROC) curve in subsequent discussions.
Overall, the confusion matrix provides valuable insights into the performance of a classifier, allowing us to assess its ability to correctly predict class labels.
12.2 Precision, Recall, and F1 Score (L12 Model Eval 5: Performance Metrics)
12.2 Precision, Recall, and F1 Score (L12 Model Eval 5: Performance Metrics)
In the previous video, we discussed the confusion matrix, which is a useful tool for evaluating classification models. It allows us to compute the number of true positives, false positives, true negatives, and false negatives. We also explored the true positive rate and true negative rate. Now, we will extend our understanding by introducing three additional metrics: precision, recall, and F1 score.
Let's start with precision. Precision is calculated by dividing the number of true positives by the sum of true positives and false positives. True positives are the instances that are correctly predicted as positive, while false positives are the instances that are incorrectly predicted as positive. In the context of spam classification, for example, true positives represent the emails correctly identified as spam, while false positives refer to non-spam emails incorrectly classified as spam. Precision measures the accuracy of positive predictions, answering the question: How many of the predicted spam emails are actually spam?
Next, we have recall, which is also known as the true positive rate. Recall is calculated by dividing the number of true positives by the sum of true positives and false negatives. True positives represent the instances correctly predicted as positive, and false negatives represent the instances incorrectly predicted as negative. Recall indicates how many of the actual positive instances were correctly identified as positive. In other words, it measures the effectiveness of a classifier in capturing positive instances.
Another important metric is the F1 score, which combines precision and recall into a single value. It is computed by taking the harmonic mean of precision and recall, weighted by a factor of two. The F1 score provides a balanced measure of a classifier's performance, considering both precision and recall. It is especially useful when we want to evaluate a model that performs well in both precision and recall.
All of these metrics have a range between zero and one, with one being the best possible value. In terms of terminology, sensitivity and specificity are more commonly used in computational biology, while precision and recall are popular in information technology, computer science, and machine learning. When choosing which metrics to use in a paper or study, it's important to consider the conventions of the specific field.
To better understand precision and recall, let's visualize them using a helpful diagram from Wikipedia. In this visualization, we consider the positive class (e.g., spam emails) as everything on the left and the negative class as everything on the right. Precision is represented by the true positives divided by all the predicted positives, while recall is represented by the true positives divided by all the actual positives.
Additionally, we have two other commonly used metrics: sensitivity and specificity. Sensitivity is another term for recall, representing the true positive rate. Specificity, on the other hand, is the number of true negatives divided by the number of negatives. It complements sensitivity and focuses on the accurate identification of negative instances.
Now, let's discuss the Matthews correlation coefficient. Initially designed for assessing protein secondary structure predictions in biology, this coefficient measures the correlation between true and predicted labels. It can be considered as a binary classification counterpart to Pearson's correlation coefficient. Similar to Pearson's r, Matthews correlation coefficient ranges from -1 to 1, with 1 indicating a perfect match between the true and predicted labels. It is particularly useful in imbalanced class problems, where one class has significantly more examples than the other.
To calculate these metrics, you can use functions provided by scikit-learn, such as precision_score, recall_score, f1_score, and matthews_corrcoef. These functions take the true labels and predicted labels as inputs and return the corresponding metric values. Alternatively, you can use these metrics in grid search and hyperparameter tuning, providing the desired metric as a string argument.
For multi-class problems, when you want to use metrics like precision, recall, F1 score, or Matthews correlation coefficient, you need to apply a workaround. One approach is to use the make_scorer function from scikit-learn. This function allows you to create a scoring object for a specific metric.
For example, if you want to use the F1 score for a multi-class problem, you can create a scorer object using make_scorer and set the average parameter to "macro" or "micro". The "macro" option calculates the metric independently for each class and then takes the average, while the "micro" option considers the total number of true positives, false negatives, and false positives across all classes.
It's important to note that the choice between "macro" and "micro" averaging depends on the problem and the specific requirements of your analysis.
In addition to using these metrics individually, you can also apply them in grid search and hyperparameter tuning. Instead of using the classification accuracy as the scoring metric, you can provide the desired metric as a string argument in the grid search process. This allows you to optimize your model based on the chosen metric, providing a more comprehensive evaluation than just relying on accuracy.
Remember that when working with multi-class problems and applying these metrics, it's crucial to understand the averaging options and choose the appropriate method based on your specific needs.
In summary, in this video, we covered additional evaluation metrics for classification models, including precision, recall, F1 score, and Matthews correlation coefficient. These metrics provide valuable insights into the performance of a classifier, considering factors such as true positives, false positives, and false negatives. By using these metrics, you can gain a deeper understanding of how well your model is performing and make informed decisions in your analysis or research. In the next video, we will delve into balanced accuracy, receiver operating characteristic (ROC) curves, and extending binary metrics to multi-class settings, expanding our knowledge of evaluation techniques in classification tasks.
12.3 Balanced Accuracy (L12 Model Eval 5: Performance Metrics)
12.3 Balanced Accuracy (L12 Model Eval 5: Performance Metrics)
Alright, let's now delve into the concept of balanced accuracy, which is particularly helpful when dealing with class imbalance problems in classification tasks. Class imbalance occurs when one class has a significantly larger number of labels than another class. To illustrate this, let's consider a multi-class classification problem using the example of Iris flowers, specifically Iris setosa, Iris versicolor, and Iris virginica.
Typically, we compute prediction accuracy by summing the values on the diagonal of the confusion matrix and dividing it by the total number of examples. In the given example, we have 3 labels for Class Zero, 769 labels for Class One, and 18 labels for Class Two. As you can see, there is an imbalance in the classes, with Class One having a higher number of examples compared to the other two classes. If we calculate the regular accuracy, it would be around 80%, primarily influenced by the high number of Class One examples.
However, the regular accuracy may not provide an accurate representation of the model's performance, especially when the focus should be on achieving balanced predictions for all classes. In such cases, the balanced accuracy metric aims to provide a more equitable evaluation by giving equal weight to each class.
To compute the balanced accuracy, we consider each class as the positive class and merge the remaining classes into the negative class. For instance, let's focus on Class Zero first. We treat Class Zero as the positive class and combine Class One and Class Two as the negative class. By analyzing the confusion matrix, we can determine the true positives, true negatives, false positives, and false negatives for Class Zero. This process is repeated for each class, creating separate binary classification problems.
In Python, you can use the accuracy_score function from the mlxtend library to compute the balanced accuracy. This function operates similarly to scikit-learn's accuracy_score but includes additional functionality to calculate the binary classification accuracy. By specifying the method as binary and providing the positive label, you can compute the binary accuracy for each class. Additionally, you can use the average parameter as "macro" or "micro" to calculate the average per-class accuracy directly.
In the example provided, the confusion matrix is recreated, and the regular accuracy, average per-class accuracy (balanced accuracy), and binary accuracies are computed. The binary accuracies correspond to each class treated as the positive label separately. By averaging the binary accuracies, you obtain the balanced accuracy. In this case, the balanced accuracy is approximately 86%.
Balanced accuracy, or average per-class accuracy, provides a fair evaluation of a classifier's performance in multi-class problems with class imbalance. It considers each class equally and offers insights into the model's ability to predict all classes accurately.
Moving on, let's now discuss the Receiver Operating Characteristic (ROC) curve, another important evaluation metric in machine learning. The ROC curve is a graphical representation of the performance of a binary classifier, and it provides valuable insights into the trade-off between the true positive rate (TPR) and the false positive rate (FPR) at different classification thresholds.
To understand the ROC curve, let's first define the true positive rate (TPR) and the false positive rate (FPR). TPR, also known as sensitivity or recall, measures the proportion of actual positive instances that are correctly identified by the classifier. It is calculated as the number of true positives divided by the sum of true positives and false negatives:
TPR = True Positives / (True Positives + False Negatives)
On the other hand, the false positive rate (FPR) measures the proportion of actual negative instances that are incorrectly classified as positive. It is calculated as the number of false positives divided by the sum of false positives and true negatives:
FPR = False Positives / (False Positives + True Negatives)
To construct the ROC curve, the classifier's predictions are ranked according to their classification scores or probabilities. By varying the classification threshold, we can generate different TPR and FPR values. Starting from the threshold that classifies all instances as positive (resulting in a TPR of 1 and an FPR of 1), we gradually decrease the threshold, classifying fewer instances as positive, and consequently, reducing both the TPR and FPR.
Plotting the TPR against the FPR for each threshold value gives us the ROC curve. The curve illustrates the classifier's performance across various operating points, with the ideal scenario being a curve that hugs the top-left corner, indicating high TPR and low FPR for all threshold values.
In addition to the ROC curve, another important metric derived from it is the Area Under the ROC Curve (AUC-ROC). The AUC-ROC quantifies the overall performance of the classifier by calculating the area under the ROC curve. A perfect classifier has an AUC-ROC of 1, indicating that it achieves a TPR of 1 while maintaining an FPR of 0. Conversely, a random classifier has an AUC-ROC of 0.5, as it performs no better than chance.
The ROC curve and AUC-ROC provide a comprehensive analysis of a binary classifier's performance, irrespective of the chosen classification threshold. It allows us to compare different classifiers or different settings of the same classifier, enabling informed decisions about model selection.
To compute the ROC curve and AUC-ROC in Python, various libraries such as scikit-learn offer convenient functions. These functions take the true labels and the predicted probabilities or scores as inputs and return the FPR, TPR, and thresholds for the ROC curve, as well as the AUC-ROC value.
In summary, the ROC curve and AUC-ROC are valuable tools for evaluating and comparing the performance of binary classifiers. They provide insights into the trade-off between true positive and false positive rates at different classification thresholds, allowing for informed decision-making in model selection.
12.4 Receiver Operating Characteristic (L12 Model Eval 5: Performance Metrics)
12.4 Receiver Operating Characteristic (L12 Model Eval 5: Performance Metrics)
The topic of discussion revolves around the receiver operating characteristic (ROC) curve. This curve, also known as the receiver operating characteristic (ROC) curve, may sound like a tongue twister due to its complex name. The term "receiver operating characteristic" originates from the field of radar receiver operators who worked with technologies such as radio direction ranging. Initially, it was used in this context. However, it has gained popularity in the field of machine learning due to its ability to combine two essential metrics: the true positive rate and the false positive rate.
The ROC curve is constructed by varying the prediction threshold. To illustrate this, let's consider an example. In binary classification problems, we have two classes: Class Zero and Class One. Rather than having a simple binary classification decision, we can assign a class membership probability to each example. This probability can be determined using various classifiers such as logistic regression, k-nearest neighbors, or decision trees. For instance, in the case of k-nearest neighbors, the class membership probability can be calculated as the ratio of the occurrences of one class over the other in the neighborhood.
Let's consider an example using the k-nearest neighbors algorithm. Suppose we have a set of neighbors, and we want to determine the class membership probability for a specific example. If we observe that out of the five nearest neighbors, three belong to Class Zero and two belong to Class One, the class membership probability for Class Zero would be calculated as 3/5, which is 0.6, and for Class One as 2/5, which is 0.4. These probabilities can be adjusted based on a threshold.
The threshold represents the point at which we make the decision between Class Zero and Class One. For example, if we set the threshold at 0.5, any probability above 0.5 would be classified as Class One, and any probability below 0.5 would be classified as Class Zero. In logistic regression, 0.5 is commonly used as the threshold. However, the threshold can be arbitrary and depends on our objectives, whether we want to optimize for the true positive rate, the false positive rate, or any other criteria. The choice of threshold also includes a tiebreaker rule, such as selecting the lower class in case of a tie.
The receiver operating characteristic (ROC) curve illustrates the performance of a classifier by plotting the true positive rate (TPR) on the y-axis and the false positive rate (FPR) on the x-axis. Each point on the curve corresponds to a different threshold value. By changing the threshold, we can observe how the classifier's performance is affected in terms of the true positive rate and false positive rate. The curve depicts the sensitivity of the classifier to different threshold values, enabling us to analyze its behavior comprehensively.
In terms of accuracy, we usually use a fixed threshold, such as 0.5, to determine the class assignment. However, for the receiver operating characteristic curve, we explore the classifier's sensitivity by changing the threshold. By doing so, we can assess the impact of varying thresholds on the false positive rate and true positive rate. The curve displays different points corresponding to different thresholds, such as 0.3, 0.4, 0.5, 0.6, and so on. Each point on the curve represents the classifier's performance for a specific threshold.
Now, let's delve into the concept of threshold and its relationship to the class membership probability. Previously, we saw a figure that depicted a feature on the x-axis and the decision boundary being shifted. However, this was merely an abstraction to aid in understanding. In reality, the x-axis represents the class membership probability.
So, continuing from where we left off, I will explain the code snippet further. After computing the probabilities for class one, we use the roc_curve function from scikit-learn to calculate the false positive rate (fpr) and true positive rate (tpr) for different thresholds. The roc_curve function takes the true labels (y_true) and the predicted probabilities (y_scores) as inputs and returns the fpr, tpr, and thresholds.
Next, we use the roc_auc_score function to compute the area under the receiver operating characteristic curve (AUC-ROC). This metric provides a single value that summarizes the performance of the classifier across all possible thresholds. A higher AUC-ROC indicates better classification performance. We calculate the AUC-ROC for both the training and testing sets separately.
Finally, we plot the ROC curve using Matplotlib. The plot shows the ROC curve for the training set in blue and the testing set in orange. We also add labels and a legend to the plot for better interpretation.
By visualizing the ROC curves and calculating the AUC-ROC, we can assess the performance of the classifier and compare it between the training and testing sets. If the AUC-ROC is close to 1, it indicates a good classifier with a high true positive rate and a low false positive rate. On the other hand, an AUC-ROC close to 0.5 suggests a random or ineffective classifier.
In summary, the code snippet demonstrates how to use the receiver operating characteristic (ROC) curve and the area under the curve (AUC) as evaluation metrics for a binary classification problem. The ROC curve visualizes the trade-off between the true positive rate and false positive rate at different prediction thresholds, while the AUC-ROC provides a single value to quantify the classifier's performance.